MONOTONICITY OF EIGENVALUES

LEON GREENBERG

lng@math.umd.edu

Department of Mathematics
University of Maryland
College Park
Maryland MD 20742 , USA

For $0 < y \le 1$, consider the family of Sturm-Liouville problems:

$$-(p(x)u')' + q(x)u = \lambda r(x)u, \ \ 0 < x < y, \hspace{1.5in} \ \ (1a)$$

$$\alpha _0 u(0) + \beta _0 p(0)u'(0) = 0, \hspace{2.3in} \ \ (1b)$$

$$\alpha _1 u(y) + \beta _1 p(y)u'(y) = 0, \hspace{2.3in} \ \ (1c)$$

where $p\in C^1[0,1], \ q,\ r\in C[0,1]$, and $p,\ r > 0$. The Sturm-Liouville problem (1) has a sequence of eigenvalues: $\lambda _0(y) < \lambda _1(y) < \lambda _2(y) < \cdots$. A well known Sturmian property states that if the second boundary condition (1c) is a Dirichlet condition (i.e. if $\beta _1 = 0$) then the eigenvalues $\lambda _n(y)$ are strictly decreasing functions of $y$. We ask the question: Does any vestige of this property remain true if $\beta _1 \ne 0$? The answer is yes.

Theorem 1   There exists an integer $n_0 \ge 0$ such that for $n \ge n_0$, the eigenvalues $\lambda _n(y)$ are strictly decreasing functions of $y$.

Theorem 2   For given functions $p,\ r$ and a given integer $n_0 \ge 0$, there exists a continuous function $q$ such that the eigenvalues $\lambda _n(y), \ 0 \le n \le n_0,$ are not decreasing. In fact, for any given $y_0 \in (0,1)$, $q$ can be found so that $\lambda _n(y)$ is increasing in a neighborhood of $y_0, \ 0 \le n \le n_0$.